Two dice are throw simultaneously . Find the probability that (i) both dice show the same number, (ii) the product of the numbers on the dice is 8, (iii) the total of the numbers on the dice is greater than 9.

Answer

To solve the problem step by step, let's break down each part of the question regarding the probability of different outcomes when two dice are thrown simultaneously.

### Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Part (i): Probability that both dice show the same number
1. **Identify favorable outcomes**: The outcomes where both dice show the same number are:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)

This gives us a total of 6 favorable outcomes.

2. **Calculate probability**: The probability is given by the formula:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Part (ii): Probability that the product of the numbers on the dice is 8
1. **Identify favorable outcomes**: The pairs of numbers that give a product of 8 are:
- (2, 4)
- (4, 2)

This gives us a total of 2 favorable outcomes.

2. **Calculate probability**: The probability is:
\[ \text{Probability} = \frac{2}{36} = \frac{1}{18} \]

### Part (iii): Probability that the total of the numbers on the dice is greater than 9
1. **Identify favorable outcomes**: The pairs of numbers that have a sum greater than 9 are:
- (4, 6)
- (5, 5)
- (5, 6)
- (6, 4)
- (6, 5)
- (6, 6)

This gives us a total of 6 favorable outcomes.

2. **Calculate probability**: The probability is:
\[ \text{Probability} = \frac{6}{36} = \frac{1}{6} \]

### Summary of Probabilities
- (i) Probability that both dice show the same number: \(\frac{1}{6}\)
- (ii) Probability that the product of the numbers on the dice is 8: \(\frac{1}{18}\)
- (iii) Probability that the total of the numbers on the dice is greater than 9: \(\frac{1}{6}\)